In either case, the SAFC on \(\mathcal {D}^{\Pi }_{\pi /2}\)  reduces to the flow tree formula for quivers, which is proven in [1]}. Moreover, the fact that rays do not intersect away from orbifold points
makes it clear that the index \(\Omega _c(\gamma )\)  computed using the quiver
associated to the exceptional collection
\(E_i(k)\)  will coincide with the Gieseker index
\(\Omega _\infty (\gamma )\)  along the ray \(\mathcal {R}_{\pm \pi /2}(\gamma )\)  provided
\(k-1\le \frac{d}{r}\le k\)  for \(r\ne 0\) , as observed in [2]}, [3]}.
